polarization

1

POLARIZATION

SOLO HERMELIN

Updated 27.05.06http://www.solohermelin.com

http://www.solohermelin.com/

POLARIZATION

History

SOLO

TABLE OF CONTENT

Natural or Unpolarized Light

Monochromatic Planar Wave Equations

Linear Polarization or Plane-PolarizationCircular Polarization

Elliptically PolarizationMethods of Achieving Polarization

Polarization Ellipse

Degenerated States of Polarization Ellipse

The Stokes Polarization Parameters

Measuring the Stokes Parameters

Poincaré Sphere

The Mueller Matrices for Polarizing Components

The Jones Polarization Parameters

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POLARIZATIONSOLO

TABLE OF CONTENT (Continue – 1)

Faraday Effect

Electro – Optical Effects ( Pockels, Kerr)

References Optics Polarization

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POLARIZATION

Erasmus Bartholinus, doctor of medicine and professor of mathematics at the University of Copenhagen, showed in 1669 that crystals of “Iceland spar” (which we now call calcite, CaCO3) produced two refracted rays from a single incident beam. One ray, the “ordinary ray”, followed Snell’s law, while the other, the “extraordinary ray”, was not always even in the plan of incidence.

SOLO

History

Erasmus Bartholinus1625-1698

http://www.polarization.com/history/history.html

a, 05/12/2006

M.V.Klein, T.Furtak,"Optics", pp.34-35E. Hecht, A. Zajac,"Optics",pp.8, pp.225-226W.Swindell,Ed.,"Polrized Light", BenchmarkPapers in Optics, V.1, pg.10 and 25

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POLARIZATIONSOLO

History

Étienne Louis Malus1775-1812

Etienne Louis Malus, military engineer and captain in the army ofNapoleon, published in 1809 the Malus Law of irradiance through aLinear polarizer: I(θ)=I(0) cos2θ. In 1810 he won the French AcademyPrize with the discovery that reflected and scattered light also possessed“sidedness” which he called “polarization”.

a, 08/20/2005

M.V.Klein, T.Furtak,"Optics", pp.34-35E. Hecht, A. Zajac,"Optics",pp.8, pp.225-226

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POLARIZATION

Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that tworays polarized at right angles to each other never interface.

SOLO

History (continue)

Dominique François Jean Arago1786-1853

Augustin Jean Fresnel

1788-1827

Arago relayed to Thomas Young in London the resultsof the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillationsin the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young

1773-1829

Return to Table of Content

solo, 08/21/2007

M. Born, E. Wolf,"Principles of Optics", pp. xxiiiM.V. Klein, T. Furtak,"Optics", pp.35

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POLARIZATIONSOLO

The Natural Light is emitted by excitation of material atoms. Each excited atomradiates a wave train for roughly 10-8 sec. The Natural light is the result of radiation of alarge collection of such atoms. New waves are emitted in a completely unpredictable fashionand the light is actually composed of a succession of short living polarization states.

Natural or Unpolarized Light

A light source consists of a very large number of randomly oriented atoms emitters.

We say that the Natural light is compose of a collection of monochromatic (polichromatic) unpolarized rays.

One mathematical description of a monochromatic unpolarized ray moving in z direction, at a certain location in space, is by a Electric Intensity phasor of constant amplitude and a random phase:

( ) ( )( ) ( )( ) yxtzktjtzktj yx eAeAtE 11

∧+−

∧+− += δωδω

zyx 111 ,,∧∧∧

are orthogonal unit vectors and δx (t) are δy (t) are randomly phase angles.

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POLARIZATIONSOLO

Light is a transverse electromagnetic wave; i.e. the Electric and Magnetic Intensitiesare perpendicular to each other and oscillate perpendicular to the direction of propagation.

For the natural light the direction of the Electric Intensity vector changes randomlyfrom time to time. We say that the natural light is Unpolarized.

A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized.

If light is composed of two plane waves of equal amplitude but differing in phase by 90°then the light is aid to be Circular Polarized.

If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized.

E


9

ELECTROMAGNETICSSOLO

To satisfy the Maxwell equations for a source free media we must have:

Monochromatic Planar Wave Equations

we haveUsing: 1ˆˆ&ˆˆ0 =⋅== kkknkkk εµω

=⋅∇=⋅∇

−=×∇=×∇

0

0

H

E

HjE

EjH

ωµωε

=⋅

=⋅

=×

−=×

0ˆ

0ˆ

ˆ

ˆ

0

0

00

00

Hk

Ek

HEk

EHk

εµ

µε

=⋅−

=⋅−

−=×−

=×−

⇒

⋅−

⋅−

⋅−⋅−

⋅−⋅−

−=∇ ⋅−⋅−

0

0

0

0

00

00

rkj

rkj

rkjrkj

rkjrkj

ekje

eHkj

eEkj

eHjeEkj

eEjeHkjrkjrkj

ωµ

ωε

( ) *2

*2

&2

ˆ

2

ˆHHwEEwwcn

kwwcn

kS meme

µε ====+=

Time Average Poynting Vector of the Planar Wave

( ) ( )rktjrktj eHHeEE

⋅−⋅− == ωω00 &


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POLARIZATIONSOLO

A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized.

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

http://www.enzim.hu/~szia/cddemo/edemo0.htm )Andras Szilagyi(

Linear Polarization or Plane-Polarization

( ) yyzktj

y eAE 1∧

+−= δω


http://www.enzim.hu/~szia/cddemo/edemo0.htm

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POLARIZATIONSOLO


If light is composed of two plane waves of equal amplitude but differing in phase by 90°then the light is said to be Circular Polarized.

http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm

( ) ( ) yx xx zktjzktj eAeAE 11 2/∧

++−∧

+− += πδωδω

Circular Polarization


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POLARIZATIONSOLO


If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized.

( ) ( ) yx yxzktj

y

zktj

x eAeAE 11∧

+−∧+− += δωδω

Elliptically Polarization


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POLARIZATIONSOLO


Methods of Achieving Polarization

Polarization is based on one of four fundamental physical mechanisms:

1. Dichroism or selective absorbtion 3. Reflection

2. Scattering 4. Birefrigerence

To obtain Polarization we must have some asymmetry in the optical process.

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POLARIZATIONSOLO


Methods of Achieving Polarization (continue – 1)

Polarization by Dichroism

A dichroic material has different absorption properties for perpendicular incident planes. An example of a dichroic material is the tourmaline that is a class of borone silicate. The tourmaline has a unique optic axis, and any electronic field normal to it is strongly absorbed.

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POLARIZATIONSOLO

Polaroids (Dichroism)

http://en.wikipedia.org/wiki/Edwin_H._Land

Edwin H. Land 1909-1991

In 1928 Edwin H. Land undergraduate at Harvard Collegeinvented the Polaroid J-sheet. It consists of many microscopic Crystals of iodoquinine sulphate embeded in a transparentNitrocellulose polymer film.

The sunglasses use polaroid material that uses dichroism to achieve absorption..

16


Wire-Grid Polarizer (Dichroism)

POLARIZATIONSOLO

Grid of parallel conducting wires with a spacing comparable to the wavelength of the electromagnetic wave.

The Electric Field vector parallel to the wires is attenuated because of the currents induced in the wires.

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POLARIZATIONSOLO



Polarization by Scattering

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POLARIZATIONSOLO


Polarization by Scattering

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POLARIZATIONSOLO



Polarization by Reflection

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POLARIZATIONSOLO


Polarization by Reflection

The Pile-of-plate Polarizer

The problem encounter using the Brewster Effect is that the reflected beam although completely polarized is weak and the refracted beam is only partially polarized.

The solution is to use a pile-of-plates polarizer as in Figure.

This was invented by F.J. Arago in 1812.

Dominique François Jean Arago1786-1853

soloh, 05/18/2006

Hecht & Zajec, "Optics", Addison Wesley, 1979, p.245

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POLARIZATIONSOLO


Polarization by Birefrigerence (continue – 4)

Polarization can be achieved with crystalline materials which have a different index ofrefraction in different planes. Such materials are said to be birefringent or doubly refracting.

Nicol Prism The Nicol Prism (1828) is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray.

William Nicol(1768 ?– 1851) Scottish physicist


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POLARIZATIONSOLO




Wollaston Prism

William HydeWollaston1766-1828


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POLARIZATIONSOLO




Glan-Foucault Polarizer


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POLARIZATION


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POLARIZATION


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POLARIZATIONSOLO

Polarizations Prisms Overview

http://www.unitedcrystals.com/POverview.html


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POLARIZATIONSOLO

Polarizations Prisms Overview

http://www.unitedcrystals.com/POverview.html



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POLARIZATIONSOLO

The Polarization of a monochromatic planar wave is defined in terms of the behavior of the tip of the phasor vector as a function of timeE

( ) ( ) yx yxzktj

y

zktj

x eAeAE 11∧

+−∧+− += δωδω

Tacking the real part we obtain:( )x

x

x zktA

E δω +−= cos

( ) ( ) δδωδδωδδδωδ

sinsincoscoscos xxxyx

y

y zktzktzktA

E+−−+−=

−++−=

δδ sin1cos

2

−=

x

x

x

x

y

y

A

E

A

E

A

E

δδ 2

22

sin1cos

−=

−

x

x

x

x

y

y

A

E

A

E

A

E

δδ 2

22

sincos2 =

+−

y

y

y

y

x

x

x

x

A

E

A

E

A

E

A

E

1sinsin

cos2

sin

2

2

2

=

+−

δδ

δδ y

y

y

y

x

x

x

x

A

E

A

E

A

E

A

E ellipse

( )y

y

y zktA

Eδω +−= cos

Polarization Ellipse

29

POLARIZATIONSOLO

Let transform the ellipse equation to canonical form by using a linear transformationthrough an angle ψ (to be defined)

1sinsin

cos2

sin

2

2

2

=

+−

δδ

δδ y

y

y

y

x

x

x

x

A

E

A

E

A

E

A

E

−=

⇔

−

=

η

ξ

η

ξ

ψψψψ

ψψψψ

E

E

E

E

E

E

E

E

y

x

y

x

cossin

sincos

cossin

sincos

1sin

cossincos

sin

cossin

sin

sincos2

sin

sincos22

=

++

+

−−

−δ

ψψδ

δψψ

δψψ

δψψ ηξηξηξηξ

yyxx A

EE

A

EE

A

EE

A

EE

( ) 1sincossin

cos2

sin

cossin2

sin

cossin2

cossin

cos

sin

sin2

sin

cos

sin

sincos

sin

sin

sin

cos2

sin

sin

sin

cos

0

22

22222

22

2

22

22

22

2

22

22

=

−−+−+

+++

−+

ψ

ηξ

ηξ

ψψδ

δδψψ

δψψ

δδ

ψδ

ψδ

ψδ

ψδδ

ψδ

ψδ

ψδ

ψ

definingby

yxyx

yxyxyxyx

AAAAEE

AAAAE

AAAAE

Choose ψ such that the last term is zero, or

02coscos211

2sin22

=−

− ψδψ

yxxyAAAA ( ) δαδδψ

α

cos2tancos/1

/cos

22tan

/:tan

222

xy AA

xy

xy

yx

yx

AA

AA

AA

AA =

=−

=−

=

Polarization Ellipse (continue – 1)

30

POLARIZATIONSOLO

1

22

=

+

η

η

ξ

ξ

A

E

A

E

Define (see Figure)

δδ

ψδ

ψδ

ψδ

ψ

δδ

ψδ

ψδ

ψδ

ψ

χξ

η

cossin

cos

sin

sin2

sin

cos

sin

sin

cossin

sin

sin

cos2

sin

sin

sin

cos

:tan

22

2

22

2

22

2

22

2

2

2

yxyx

yxyx

AAAA

AAAA

A

A

++

−+=

=

( ) ( )( ) ( )

δψψαψψαδψψαψψα

δψψψψδψψψψ

χ

α

coscossintan2cossintan

coscossintan2sincostan

coscossin/2cossin/

coscossin/2sincos/tan

222

222tan/

222

222

2

++−+

++−+

=

==xy EE

xyxy

xyxy

EEEE

EEEE

( ) ( ) ( )α

δψααψα

δψψααψψχχχ

2

2

2

222

2

2

tan1

cos2sintan2tan12cos

tan1

coscossintan4tan1sincos

tan1

tan12cos

++−=

++−−=

+−=

From δαψ cos2tan2tan =αψδ2tan

2tancos =

andααα

ααα

2

2

2 tan1

tan12cos&

tan1

tan22tan

+−=

−=

( )ψα

ψψψ

αα

ααψψααψ

χ

ψα

2cos

2cos

2cos

2sin2cos

tan1

tan1

tan12tan2tan

2sintan2tan12cos2cos

2cos/1

2

2cos

2

2

2

2

=

+

+−=

+

+−=

δαψ cos2tan2tan =


31

POLARIZATIONSOLO

1

22

=

+

η

η

ξ

ξ

A

E

A

E

χαψ2cos

2cos2cos = δαψ cos2tan2tan =

χδαδα

χαψψψ

2cos

cos2sincos2tan

2cos

2cos2tan2cos2sin ===

Therefore

1cossin

cos

sin

sin2

sin

cos

sin

sincos

sin

sin

sin

cos2

sin

sin

sin

cos22

2

22

22

22

2

22

22 =

+++

−+ δ

δψ

δψ

δψ

δψδ

δψ

δψ

δψ

δψ

ηξyxyxyxyx

AAAAE

AAAAE

Also

δδ

ψδ

ψδ

ψδ

ψ

δδ

ψδ

ψδ

ψδ

ψ

η

ξ

cossin

cos

sin

sin2

sin

cos

sin

sin1

cossin

sin

sin

cos2

sin

sin

sin

cos1

22

2

22

2

2

22

2

22

2

2

yxyx

yxyx

AAAAA

AAAAA

++=

−+=

δηξ22222 sin

11111

+=+

yx AAAA


To Stokes Parameters

32

POLARIZATIONSOLO

Let compute the area of the Polarization Ellipse

( )

( )yyyyy

xxxxx

AzktAEy

AzktAEx

δτδω

δτδω

τ

τ

+=

+−==

+=

+−==

coscos:

coscos:

( ) ( )

( ) ( ) δπτδδτδδ

τδτδτ

π

δ

π

sin2sinsin2

1

sincos

2

0

2

0

yxyxxyyx

xxyy

AAdAA

dAAdxyArea

=

++−−=

++−==

∫

∫∫

But the area of the Polarization Ellipse is1

22

=

+

η

η

ξ

ξ

A

E

A

Eηξπ AAArea =

Therefore δηξ sinyx AAAA =

Using

δηξ22222 sin

11111

+=+

yx AAAA δηξ

ηξ222

22

22

22

sin

1

yx

yx

AA

AA

AA

AA +=

+ 2222

yx AAAA +=+ ηξ

Energy Equation


33

POLARIZATIONSOLO

δηξ sinyx AAAA =2222

yx AAAA +=+ ηξ Energy Equation

ξ

ηχA

A=:tanwe defined

Therefore

δαδδ

χχχ

ηξ

ηξ

ξ

η

ξ

η

sin2sinsin

1

2sin22

1

2

tan1

tan22sin

2222222=

+

=+

=+

=

+

=+

=

y

x

y

x

yx

yx

A

A

A

A

AA

AA

AA

AA

A

A

A

A

δαχ sin2sin2sin =


δαψ cos2tan2tan =We also found that

ψχ

ψχψχ

αψχ

αψαχ

δδδ

2sin

2tan

2cos2cos2tan

2sin

2cos2tan

2sin

2tan/2tan

2sin/2sin

cos

sintan =

⋅⋅=

⋅===

ψχδ2sin

2tantan =

ψχα 2cos2cos2cos ⋅=

To StokesParameters

34

POLARIZATIONSOLO

Summary


δηξ sinyx AAAA =2222

yx AAAA +=+ ηξ

==

δαχδαψ

sin2sin2sin

cos2tan2tan

=

⋅=

ψχδ

ψχα

2sin

2tantan

2cos2cos2cos

⇐

χψ

δα

⇐

δα

χψ

ξ

ηχA

A=:tan

x

y

A

A=:tanα

1sinsin

cos2

sin

2

2

2

=

+−

δδ

δδ y

y

y

y

x

x

x

x

A

E

A

E

A

E

A

E1

22

=

+

η

η

ξ

ξ

A

E

A

E

−=

η

ξ

ψψψψ

E

E

E

E

y

x

cossin

sincos

xy δδδ −=( ) ( ) yxyx yxzktj

y

zktj

xyx eAeAEEE 1111∧

+−∧+−

∧∧

+=+= δωδω


35

POLARIZATIONSOLO

( ) ( ) yx yxzktj

yzktj

x eAeAE 11∧

+−∧

+− += δωδω

Linearly Horizontally Polarized (LHP):

( )x

x

x zktA

E δω +−= cos


( ) 01 ==∧

+−y

zktjx AeAE xxδω

http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)

Linearly Vertically Polarized (LVP): ( )01 ==

∧+−

xzktj

y AeAE yyδω

( )y

y

y zktA

Eδω +−= cos

36

POLARIZATIONSOLO

Linear + 45° Polarized (L+45P)


( )tzkj

yx eAAE yx ω−−∧∧

+= 11 ( )tzkj

yx eAAE yx ω−−∧∧

−= 11

Plane PolarizationMixed

0=−= xy δδδ πδδδ =−= xy

Linear - 45 ° Polarized (L-45P)


( ) ( ) yx yxzktj

yzktj

x eAeAE 11∧

+−∧

+− += δωδω

37

POLARIZATIONSOLO

( )xx zkt

A

E δω +−= cos ( )x

y zktA

Eδω +−−= sin


Right Circular Polarization (RCP) AAA yxxy ===−= &2/πδδδ

1

22

=

+

A

E

A

E yx


( ) ( ) yx yxzktj

yzktj

x eAeAE 11∧

+−∧

+− += δωδω

( )xx zkt

A


y zktA

Eδω +−= sin

Left Circular Polarization (LCP) AAA yxxy ===−= &2

3πδδδ

38

POLARIZATIONSOLO

( )xx zkt

A


y zktA

Eδω +−= sin


Left Circular Polarization (LCP) AAA yxxy ===−= &2

3πδδδ

1

22

=

+

A

E

A

E yx


Superposition of Two Circular Polarizations

( ) ( ) yx yxzktj

yzktj

x eAeAE 11∧

+−∧

+− += δωδω


a, 05/13/2006

39

POLARIZATIONSOLOThe Stokes Polarization Parameters

George Gabriel Stokes 1819-1903

G.G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from different Sources”,

Trans. Cambridge Phil. Soc., Vol.9, 1852, pp.399-416

( ) ( ) yx yyxx zktAzktAE 11 coscos∧∧

+−++−= δωδω

where

( ) ( ) ( ) ( )δδ 2

22

sin,

cos,,

2,

=

+−

y

y

y

y

x

x

x

x

A

tzE

A

tzE

A

tzE

A

tzEThe Polarization Ellipse is

All the information of polarization is contained in this equation.

In order to observe the quantities involved let take the time average <…> of the time dependent quantities in the Polarization Ellipse equation.

( ) ( ) ( ) ( )δδ 2

2

2

2

2

sin,

cos,,

2,

=+−y

y

yx

yx

x

x

A

tzE

AA

tzEtzE

A

tzE

( ) ( ) ( ) ( ) yxjidttzEtzET

tzEtzET

jiT

ji ,,,,1

:,,0

lim == ∫→∞

( ) ( ) ( ) ( ) ( ) 22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+−

40

POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 1)

We obtain

( ) ( ) ( ) ( ) ( ) 22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+−

( ) ( ) ( ) ( )

( ) ( ) δδδωδδ

δωδω

δ

cos2

22coscos1

2

coscos1

,,

0

0

lim

lim

yxT

xyyxT

yx

T

yxyxT

zx

AAdtzkt

T

AA

dtzktzktAAT

tzEtzE

=

++−−−=

+−+−=

∫

∫

→∞

→∞

( ) ( ) ( )[ ]2

2cos11

2cos

1,

2

0

2

0

222

limlim xT

xT

xT

xxT

x

Adtzkt

T

AdtzktA

TtzE =+−−=+−= ∫∫

→∞→∞δωδω

( ) ( )2

cos1

,2

0

222

limy

T

yyT

y

AdtzktA

TtzE =+−= ∫

→∞δω

( ) ( ) 222222 sin22cos22 δδ yxyxyxyx AAAAAAAA =+−

By adding and subtracting on the left side of this equation we obtain

( ) ( ) ( ) ( ) 22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA =−−−+

442 yx AA +

41


The Stokes Parameters are defined as

( ) ( ) ( ) ( ) 22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA ++−=+

δδ

sin2

cos2

3

2

221

220

yx

yx

yx

yx

AAVS

AAUS

AAQS

AAIS

==

==

−==

+==

2

3

2

2

2

1

2

0 SSSS ++=

The Stokes Vector is defined as

−

+

=

=

=

δδ

sin2

cos2

22

22

3

2

1

0

yx

yx

yx

yx

AA

AA

AA

AA

V

U

Q

I

S

S

S

S

S

The Stokes Parameters are observable using aproper experiment.

42


Consider a quasi-monochromatic wave of mean frequency ω propagating in z direction

( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx

ttzkj

y

ttzkj

xxy 1111 ,,

∧∧∧+−−

∧+−− +=+= δωδω

For monochromatic waves Ax, Ay, δx, δy, ω are constant.

Quasi-monochromatic Light

For quasi-monochromatic waves Ax, Ay, δx, δy , ω are slowly changing with time.

We have

( ) ( ) ( ) ( ) yxjidttzEtzET

tzEtzET

jiT

ji ,,,,1

:,,0

lim == ∫ ∗

→∞

∗

( ) ( ) 2

0

21:,, lim x

T

xT

xx AdtAT

tzEtzE == ∫→∞

∗

( ) ( ) 2

0

21:,, lim y

T

yT

yy AdtAT

tzEtzE == ∫→∞

∗

( ) ( ) ( )yxyxj

yx

Tj

y

j

xT

yx eAAdteAeAT

tzEtzE δδδδ −−

→∞

∗ == ∫0

1:,, lim

( ) ( ) ( )yxxy j

yx

Tj

x

j

yT

xy eAAdteAeAT

tzEtzE δδδδ −−−

→∞

∗ == ∫0

1:,, lim

43


( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx

ttzkj

y

ttzkj

xxy 1111 ,,

∧∧∧+−−

∧+−− +=+= δωδω

Quasi-monochromatic Light


( )( )

( )( ) ∗∗

∗∗

∗∗

∗∗

−=−=

+=−=

−=−=

+=+=

xyyxxyyx

xyyxxyyx

yyxxyx

yyxxyx

EEEEjAAS

EEEEAAS

EEEEAAS

EEEEAAS

δδ

δδ

sin2

cos2

3

2

22

1

22

0

( ) ( ) 2

0

21:,, lim x

T

xT

xx AdtAT

tzEtzE == ∫→∞

∗

( ) ( ) 2

0

21:,, lim y

T

yT

yy AdtAT

tzEtzE == ∫→∞

∗

( ) ( ) ( )yxyxj

yx

Tj

y

j

xT

yx eAAdteAeAT

tzEtzE δδδδ −−

→∞

∗ == ∫0

1:,, lim

( ) ( ) ( )yxxy j

yx

Tj

x

j

yT

xy eAAdteAeAT

tzEtzE δδδδ −−−

→∞

∗ == ∫0

1:,, lim

44


Stokes Vector for Different Polarization Types

LHP

=

0

0

1

1

0IS

2

0

0

x

y

AI

A

=

=

LVP

2

0

0

y

x

AI

A

=

=

−

=

0

0

1

1

0IS

=

0

1

0

1

0IS

L+45P

2

0 2

0

x

yx

AI

AA

=

==δ

−=

0

1

0

1

0IS

L-45P

2

0 2 x

yx

AI

AA

=

== πδ

=

1

0

0

1

0IS

RCP

2

0 2

2

x

yx

AI

AA

=

=

= πδ

−

=

1

0

0

1

0IS

2

0 2

2

3

x

yx

AI

AA

=

=

= πδ

LCP

δδ

sin2

cos2

3

2

22

1

22

0

yx

yx

yx

yx

AAS

AAS

AAS

AAS

==

−=

+=

2

3

2

2

2

1

2

0 SSSS ++=

( ) ( ) yxyx yxzktj

y

zktj

xyx eAeAEEE 1111∧

+−∧+−

∧∧

+=+= δωδω

45



δδ

sin2

cos2

3

2

22

1

22

0

yx

yx

yx

yx

AAS

AAS

AAS

AAS

==

−=

+=

2

3

2

2

2

1

2

0 SSSS ++=

δαχ sin2sin2sin =

αψχ 2cos2cos2cos =⋅

δαψχ cos2sin2sin2cos ⋅=⋅

We found

22

22

2

2

222 tan1

tan12cos&

tan1

tan2sin

yx

yx

yx

yx

AA

AA

AA

AA

+−

=+−=

+=

+=

ααα

ααα

x

y

A

A=:tanα

0

122

22

2cos2cosS

S

AA

AA

yx

yx =+−

=⋅ ψχ

0

222

coscos2sin2sin2cos

S

S

AA

AA

yx

yx =+⋅

=⋅=⋅δ

δαψχ

0

322

sinsin2sin2sin

S

S

AA

AA

yx

yx =+⋅

==δ

δαχ

=

=

−

−

0

31

1

21

sin2

1

tan2

1

S

S

S

S

χ

ψ


46

POLARIZATIONSOLO

Consider a quasi-monochromatic wave of mean frequency ω propagating in z directioncomposed of a Unpolarized component AUP with random phases δrx and δry and aPolarized component Ax, δx , Ay, δy

( ) ( ) ( )tzkjj

yP

j

UP

j

xP

j

UPyx eeAeAeAeAEEE yxyx yyrxxr ωδδδδ −−∧∧∧∧

+++=+= 1111

Measuring the Stokes Parameters

Pass the beam through a waveplate that induces a wave retardation of φ and a polarizer with a transmission axis at an angle β relative to x axis

( )( ) ( )( ) ( )tzkjj

yP

j

UP

j

xP

j

UP eeAeAeAeAE yx yyrxxr ωϕδδϕδδ −−∧

−∧+

+++= 11

2/2/'

( )( ) ( )( )[ ] ( )tzkjj

yP

j

UP

j

xP

j

UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/

The waveplate that induces a wave retardation of φ between the phases of x and ycomponents of the polarized light but will not affect the random phase of the unpolarized light.

The polarizer will transmit only the component along the transmission axis

47

POLARIZATIONSOLOMeasuring the Stokes Parameters (continue – 1)

( )( ) ( )( )[ ] ( )tzkjj

yP

j

UP

j

xP

j

UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/

( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] zz yyrxxryyrxxrj

yP

j

UP

j

xP

j

UP

j

yP

j

UP

j

xP

j

UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos"", 2/2/2/2/∧−−−+−−−+

∧∗ +++⋅+++=⋅= ββββϕβ ϕδδϕδδϕδδϕδδ

( )( ) ( )( )[ ] ( )( ) ( )( )[ ]( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) βββ

βββ

βββ

βββ

ββββ

ϕδδϕδδϕδδ

δϕδδϕδδδ

ϕδδϕδδδϕδ

δδδϕδδϕδ

ϕδδϕδδϕδδϕδδ

22

0

2/

0

2/

2

0

2/2

0

2/

0

2

0

2/22

0

2/

00

2/22

0

2/2

2/2/2/2/

sincossin

sin2/cossin

cossincos

cossincos2/

sincossincos

++

++

++

++

++

++

++

+=

+++⋅+++

−−−−−−−

−−−+−

−−+−−+

−−−−+

−−−+−−−+

yP

jj

yPUP

j

yPxP

jj

yPUP

jj

yPUPUP

jj

UPxP

jj

UP

j

xPyP

jj

xPUPxP

jj

xPUP

jj

UPyP

jj

UP

jj

xPUPUP

j

yP

j

UP

j

xP

j

UP

j

yP

j

UP

j

xP

j

UP

AeeAAeAAeeAA

eeAAAeeAAeeA

eAAeeAAAeeAA

eeAAeeAeeAAA

eAeAeAeAeAeAeAeA

yxpxyyxr

yyyrxyrrr

xyxyrxx

xryxryxrx

yyrxxryyrxxr

The time average Poynting vector is

48

POLARIZATIONSOLO

( ) ( )( )[ ] zyP

jj

yPxPxPUP AeeAAAAcnkS 122222 sincossincos2/∧

−−− ++++= ββββ ϕδϕδ

2

2sincossin

2

2cos1sin

2

2cos1cos

2

2

βββ

ββ

ββ

=

−=

+=

( ) ( ) ( ) ( )[ ]{ }( ) ( ) ( ) ( )[ ]( ) ( )[ ]

[ ]βϕβϕβ

βϕδβϕδβ

βϕβϕβ

βϕϕϕϕβ

δδδδ

δδ

2sinsin2sincos2cos2

2sinsinsin22sincossin22cos2

2sinsin2sincos2cos2

2sinsincossincos2cos2

********

22222

22222

22222

xyyxxyyxyyxxyyxx

yPxPyPxPyPxPyPxPUP

jj

yPxP

jj

yPxPyPxPyPxPUP

jj

yPxPyPxPyPxPUP

EEEEjEEEEEEEEEEEEcnk

AAjAAAAAAAcnk

eeAAjeeAAAAAAAcnk

jejeAAAAAAAcnk

S

−+++−++=

++−+++=

−−++−+++=

++−+−+++=

−−

−

Measuring the Stokes Parameters (continue – 2)

( )( ) ( )( )[ ] ( )( ) ( )( )[ ] zz yyrxxryyrxxrj

yP

j

UP

j

xP

j

UP

j

yP

j

UP

j

xP

j

UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos 2/2/2/2/∧−−−+−−−+

∧∗ +++⋅+++=⋅= ββββ ϕδδϕδδϕδδϕδδ

[ ]βϕβϕβ 2sinsin2sincos2cos2 3210 SjSSScnk

S +++=

( )( )

( )( )xyyPxPxyyx

xyyPxPxyyx

yPxPyyxx

yPxPUPyyxx

AAEEEEjS

AAEEEES

AAEEEES

AAAEEEES

δδ

δδ

−=−=

−=+=

−=−=

++=+=

∗∗

∗∗

∗∗

∗∗

sin2

cos2

3

2

22

1

222

0

49


( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos2

, 3210 SjSSScnk

S +++=

( )( )

( )( )xyyPxPxyyx

xyyPxPxyyx

yPxPyyxx

yPxPUPyyxx

AAEEEEjS

AAEEEES

AAEEEES

AAAEEEES

δδ

δδ

−=−=

−=+=

−=−=

++=+=

∗∗

∗∗

∗∗

∗∗

sin2

cos2

3

2

22

1

222

0

The Stokes Parameters are measured by first removing the waveplate φ = 0

( ) [ ]ββϕβ 2sin2cos2

0, 210 SSScnk

S ++==

Now the polarizer is sequentially rotate to β = 0, π/4 and π/2

( ) [ ]1020,0 SS

cnkS +=== ϕβ

( ) [ ]2020,4/ SS

cnkS +=== ϕπβ

( ) [ ]1020,2/ SS

cnkS −=== ϕπβ

For the final measurement we add the waveplate with φ = π/2 and polarizer at β = π/4

( ) [ ]3022/,4/ SS

cnkS −=== πϕπβ

( ) ( )[ ]0,2/0,02

0 ==+=== ϕπβϕβ SScnk

S

( ) ( )[ ]0,2/0,02

1 ==−=== ϕπβϕβ SScnk

S

( ) 02 0,4/22

SScnk

S −=== ϕπβ

( )2/,4/22

03 πϕπβ ==−= Scnk

SS

50


( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos2

, 3210 SjSSScnk

S +++=

( )( )

( )( )xyyPxPxyyx

xyyPxPxyyx

yPxPyyxx

yPxPUPyyxx

AAEEEEjS

AAEEEES

AAEEEES

AAAEEEES

δδ

δδ

−=−=

−=+=

−=−=

++=+=

∗∗

∗∗

∗∗

∗∗

sin2

cos2

3

2

22

1

222

0

The Stokes Parameters can measure the degree of polarization of a beam.

We can see that a beam is unpolarized iff: 00 321

2

0 ===>=+= ∗∗ SSSAEEEES UPyyxx

The Degree of Polarization is defined as: 100

2

3

2

2

2

1

222

22

≤≤++

=++

+= P

S

SSS

AAA

AAP

yPxPUP

yPxP

If the beam is completely polarized then AUP = 0: 02

3

2

2

2

1

2

0 >++= SSSS


51

POLARIZATIONSOLO

H. Poincaré, “Théorie Mathématique de la Lumiere”,Gauthiers-Villars, Paris, 1892, Vol.2, Ch.12

Poincaré Sphere

Jules Henri Poincaré1854-1912

We found

03

02

01

/2sin

/2sin2cos

/2cos2cos

SS

SS

SS

==⋅=⋅

χψχψχ

We see that the normalized Stokes Parameters can be represented as a unit vector on a sphere (called Poincaré Sphere) using the Polarized Ellipse parameters χ and ψ.

52

POLARIZATIONSOLOPoincaré Sphere

J.D. Kraus, “Electromagnetics”, 4th Ed., McGraw Hill, 1992, p.607


53

POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components

In 1948 Hans Mueller, then a professor of physics at MIT devised a matrix methodthat deals with the relation of Stokes vector at the output of an optical device to theStokes vector at the input of the optical device.

=

3

2

1

0

33323130

23222120

13121110

03020100

3

2

1

0

'

'

'

'

S

S

S

S

mmmm

mmmm

mmmm

mmmm

S

S

S

S

inputoutput SMS

=

or

Hans Mueller, “The Foundation of Optics”, J. Opt. Soc. Am., 37, pg. 110 (1947) 38, pg. 661(1948)

54


Suppose that we have a few optical elements through which the ray passes .

( ) inputinputoutput SMSMMMSMMSMS

==== 12312323

or

123 MMMM =

55


Polarizer

10'

10'

≤≤=≤≤=

yyyy

xxxx

pEpE

pEpE

The Polarizer is described by two orthogonal transmission axes that are characterized, respectively, by transmission factors px and py (0 ≤ px,py ≤ 1).

( )∗∗

∗∗

∗∗

∗∗

−=

+=

−=

+=

xyyx

xyyx

yyxx

yyxx

EEEEjS

EEEES

EEEES

EEEES

3

2

1

0

( )∗∗

∗∗

∗∗

∗∗

−=

+=

−=

+=

'''''

'''''

'''''

'''''

3

2

1

0

xyyx

xyyx

yyxx

yyxx

EEEEjS

EEEES

EEEES

EEEES

+−

−+

=

3

2

1

0

2222

2222

3

2

1

0

2000

0200

00

00

2

1

'

'

'

'

S

S

S

S

pp

pp

pppp

pppp

S

S

S

S

yx

yx

yxyx

yxyx

( )

+−

−+

=

yx

yx

yxyx

yxyx

yxPOL

pp

pp

pppp

pppp

ppM

2000

0200

00

00

2

1,

2222

2222

56


Polarizer

( )22

1

:

/tan:

yx

xy

ppp

pp

+=

= −βLet define

=

3

2

1

0

2

2

22

22

3

2

1

0

2sin000

02sin00

002cos

002cos

2

1

'

'

'

'

S

S

S

S

p

p

pp

pp

S

S

S

S

ββ

ββ

( )

=

ββ

ββ

β

2sin000

02sin00

0012cos

002cos1

2,

2ppM POL

ββ

sin

cos

pp

pp

y

x

==

57


Polarizer (continue -1)

0'

10'

=≤≤=

y

xxxx

E

pEpE

Ideal Linear Polarizer py = 0 and 0 ≤ px ≤ 1.

( )

( )

+=

=

0

0

1

1

2

0000

0000

0011

0011

2

'

'

'

'

10

2

3

2

1

0

||

2

3

2

1

0

SSp

S

S

S

S

p

S

S

S

S

x

M

x

POL

Regardless the state of polarization of the input beam the output beam isLinear Horizontally Polarized (LHP)

10'

0'

≤≤==

yyyy

x

pEpE

E

Ideal Linear Polarizer px = 0 and 0 ≤ py ≤ 1.

( )

( )

−

−=

−

−

=

⊥

0

0

1

1

2

0000

0000

0011

0011

2

'

'

'

'

10

2

3

2

1

0

2

3

2

1

0

SSp

S

S

S

S

p

S

S

S

S

y

M

y

POL

Regardless the state of polarization of the input beam the output beam isLinear Vertically Polarized (LVP)

58



Crossed Polarizers

( ) ( )

=

−

−

=

⊥

0

0

0

0

22

0000

0000

0011

0011

2

0000

0000

0011

0011

2

'

'

'

'22

3

2

1

0

2

||

2

3

2

1

0

yx

M

y

M

xpp

S

S

S

S

pp

S

S

S

S

POLPOL

Regardless the state of polarization of the input beam the output beam iscompletely blocked

Crossed Polarizers are a combination of a Linear Polarizer with the transmission axes normal to the x-axis, followed by a Linear Polarizer with the transmission axes parallel to the x-axis (or vice-versa).

59



Eigenvalues and Eigenvectors of the Linear Polarizer Mueller Matrix

Let find the Input Stokes Vectors that are unaffected by the Polarizer; i.e.:

=

+−

−+

=

3

2

1

0

3

2

1

0

2222

2222

3

2

1

0

2000

0200

00

00

2

1

'

'

'

'

S

S

S

S

S

S

S

S

pp

pp

pppp

pppp

S

S

S

S

POLM

yx

yx

yxyx

yxyx

λ

( ) 0

22000

02200

002

002

2

1,

2222

2222

44 =

−−

−+−

−−+

=−

λλ

λ

λ

λ

yx

yx

yxyx

yxyx

xyxPOL

pp

pp

pppp

pppp

IppM

==

0

0

1

1

& 1

2

1 Spx

λ

−

==

0

0

1

1

& 2

2

2 Sp y

λ

1st eigenvalue-eigenvector

2nd eigenvalue-eigenvector

We can see that only LHP and LVP are unaffected by the Linear Polarizer.

60


Waveplate

y

j

y

x

j

x

EeE

EeE2/

2/

'

'ϕ

ϕ

−==

The Waveplate is a polarizing element that introduced a phase shift φ between the orthogonal components of an optical beam. A Waveplate is a phase-shifter but is also called a retarder or a compensator.

( )∗∗

∗∗

∗∗

∗∗

−=

+=

−=

+=

xyyx

xyyx

yyxx

yyxx

EEEEjS

EEEES

EEEES

EEEES

3

2

1

0

( )∗∗

∗∗

∗∗

∗∗

−=

+=

−=

+=

'''''

'''''

'''''

'''''

3

2

1

0

xyyx

xyyx

yyxx

yyxx

EEEEjS

EEEES

EEEES

EEEES

−=

3

2

1

0

3

2

1

0

cossin00

sincos00

0010

0001

'

'

'

'

S

S

S

S

S

S

S

S

ϕϕϕϕ

( )

−=

ϕϕϕϕ

ϕ

cossin00

sincos00

0010

0001

WPM

61


Waveplate (continue – 1)Quarter-Waveplate φ = π/2

The Quarter-Waveplate transforms Linearly Polarized (L+45P or L-45P) toRight or Left Circularly Polarized (RCP or LCP) or vice-versa.

( )

−==

0100

1000

0010

0001

2/πϕWPM

( )

RCPPLPL

WPM

=

−=

=

++

1

0

0

1

0

1

0

1

0100

1000

0010

0001

0

1

0

1

2/

4545

πϕ ( )

LCPPLPL

WPM

−

=

−

−=

−=

−−

1

0

0

1

0

1

0

1

0100

1000

0010

0001

0

1

0

1

2/

4545

πϕ

( )

PLRCPRCP

WPM

45

0

1

0

1

1

0

0

1

0100

1000

0010

0001

1

0

0

1

2/

−

−=

−=

= πϕ ( )

LCP

PLLCPLCP

WPM

45

0

1

0

1

1

0

0

1

0100

1000

0010

0001

1

0

0

1

2/

+

=

−

−=

−

= πϕ

62


Waveplate (continue – 2)Half-Waveplate φ = π

( )

−−

==

1000

0100

0010

0001

πϕWPM

−−

=

−−

=

3

2

1

0

3

2

1

0

3

2

1

0

1000

0100

0010

0001

'

'

'

'

S

S

S

S

S

S

S

S

S

S

S

S

χ

ψχ

ψχ

2sin

2sin2cos

2cos2cos

0

3

0

2

0

1

=

⋅=

⋅=

S

S

S

S

S

S

=

=

−

−

0

31

1

21

sin2

1

tan2

1

S

S

S

S

χ

ψ

2'

2'

πχχ

ψπψ

−=

−=

The Half-Waveplate reverses the orientation and ellipticity of the polarization ellipse (polarization state).

63


Waveplate (continue – 3)

The phases of two adjacent Half-Waveplates add.

( ) ( )( ) ( )

+++−+

=

−

−=

3

2

1

0

2121

2121

3

2

1

0

22

22

11

11

3

2

1

0

cossin00

sincos00

0010

0001

cossin00

sincos00

0010

0001

cossin00

sincos00

0010

0001

'

'

'

'

S

S

S

S

S

S

S

S

S

S

S

S

ϕϕϕϕϕϕϕϕ

ϕϕϕϕ

ϕϕϕϕ

Eigenvalues and Eigenvectors of the Waveplate Mueller Matrix

( ) 0

cossin00

sincos00

0010

0001

4 =

−−−

−−

=−

λϕϕϕλϕ

λλ

λϕ IMWP

LHP

S

==

0

0

1

1

&1 11

λ

LVP

S

−

=−=

0

0

1

1

&1 22

λ

1st eigenvalue-eigenvector

2nd eigenvalue-eigenvector

We can see that only LHP and LVP are unaffected by the Waveplate Polarizer.

64


The Mueller Matrix of a Rotator (Coordinate Rotation)

( )( ) θθθγ

θθθγcossinsin'

sincoscos'

yxy

yxx

EEEE

EEEE

+−=−=+=−=

Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The orthogonal axes are defined as (‘).

( )∗∗

∗∗

∗∗

∗∗

−=

+=

−=

+=

xyyx

xyyx

yyxx

yyxx

EEEEjS

EEEES

EEEES

EEEES

3

2

1

0

( )∗∗

∗∗

∗∗

∗∗

−=

+=

−=

+=

'''''

'''''

'''''

'''''

3

2

1

0

xyyx

xyyx

yyxx

yyxx

EEEEjS

EEEES

EEEES

EEEES

−=

3

2

1

0

3

2

1

0

1000

02cos2sin0

02sin2cos0

0001

'

'

'

'

S

S

S

S

S

S

S

S

θθθθ ( )

−=

1000

02cos2sin0

02sin2cos0

0001

θθθθ

θROTM

'1'1

1111

''

sincos

yx

yxyx

yx

yx

EE

EEEE

∧∧

∧∧∧∧

+=

+=+= γγ

65


Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The Mueller Rotation Matrix from x,y to x’,y’ is

( )

−=

1000

02cos2sin0

02sin2cos0

0001

θθθθ

θROTM

The Mueller Polarizator Matrix is

( )

=

ββ

ββ

β

2sin000

02sin00

0012cos

002cos1

2,

2ppM POL

( ) SMS ROT

θ='

( ) ( ) ( ) SMpMSpMS ROTPOLPOL

θββ ,'," ==

The Mueller Rotation Matrix from x’,y’ to x,y is

( )

−

=−

1000

02cos2sin0

02sin2cos0

0001

θθθθ

θROTM ( ) ( ) ( ) ( ) SMpMMSMS ROTPOLROTROT

θβθθ ,"'" −=−=

( ) ( ) ( ) ( )θβθθβ ROTPOLROTPOL MpMMpM ,,, −=

The Mueller Matrix of a Rotated Polarizer

66


( ) ( )( )

+−

−+=

βθβθθθβθβ

θθβθβθθβθβθβ

θβ

2sin000

02cos2sin2sin2cos2sin2sin12sin2cos

02cos2sin2sin12sin2sin2cos2cos2cos

02sin2cos2cos2cos1

2,,

22

222ppM POL

The Mueller Matrix for Rotated Polarizer is

( ) ( ) ( ) ( )

−

−

=

−=

1000

02cos2sin0

02sin2cos0

0001

2sin000

02sin00

0012cos

002cos1

1000

02cos2sin0

02sin2cos0

0001

2

,,,

2

θθθθ

ββ

ββ

θθθθ

θβθθβ

p

MpMMpM ROTPOLROTPOL

The Mueller Matrix of a Rotated Polarizer (continue – 1)

67


( ) ( )( )

−+−

−−+=

ϕϕβϕβϕβϕββϕββϕβϕββϕββ

ϕβ

cossin2cossin2sin0

sin2coscos2cos2sincos12cos2sin0

sin2sincos12cos2sincos2sin2cos0

0001

,22

22

WPM

The Mueller Matrix for Rotated Waveplate is

( ) ( ) ( ) ( )

−

−

−

=

−=

1000

02cos2sin0

02sin2cos0

0001

cossin00

sincos00

0010

0001

1000

02cos2sin0

02sin2cos0

0001

,,

θθθθ

ϕϕϕϕθθ

θθ

θϕθθβ ROTWPROTPOL MMMpM

The Mueller Matrix of a Rotated Waveplate


68

POLARIZATIONSOLOThe Jones Polarization Parameters

R. Clark Jones, “A New Calculus for the Treatment of Optical Systems”,J. Opt. Soc. Am., Vol.31, July 1941, pp.500-503J. Opt. Soc. Am., Vol.32, Aug. 1942, pp.486-493J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.107-110J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.110-112J. Opt. Soc. Am., Vol.38, Aug. 1948, pp.671-585 R. Clark Jones

1916-2004

Mueller matrices deal with the intensity of the beam. If the phase information is important we must use Jones formalism. Jones calculus was developed in the same time with the Mueller calculus by R. Clark Jones who introduced Jones vectors and Jones matrices:

Jones vectors describe the polarization of light:

Jones matrices describe the optical component:

+=

y

x

yxE

E

EEJ

22

1

[ ]

=

2221

1211

jj

jjJ

Jones calculus deals only with polarized light.

69

POLARIZATIONSOLO

Stokes and Jones Vector for Different Polarization Types

LHP

=

=0

1

0

0

1

1

0 JIS

2

0

0

x

y

AI

A

=

=

LVP

2

0

0

y

x

AI

A

=

=

=

−

=1

0

0

0

1

1

0 JIS

=

=1

1

2

1

0

1

0

1

0 JIS

L+45P

2

0 2

0

x

yx

AI

AA

=

==δ

−

=

−=

1

1

2

1

0

1

0

1

0 JIS

L-45P

2

0 2 x

yx

AI

AA

=

== πδ

=

=i

JIS1

2

1

1

0

0

1

0

RCP

2

0 2

2

x

yx

AI

AA

=

=

= πδ

−

=

−

=i

JIS1

2

1

1

0

0

1

0

2

0 2

2

3

x

yx

AI

AA

=

=

= πδ

LCP

δδ

sin2

cos2

3

2

22

1

22

0

yx

yx

yx

yx

AAS

AAS

AAS

AAS

==

−=

+=

2

3

2

2

2

1

2

0 SSSS ++=

( ) ( ) yxyx yxzktj

y

zktj

xyx eAeAEEE 1111∧

+−∧+−

∧∧

+=+= δωδω


70

POLARIZATIONSOLO

Faraday Effect (Hecht p.261) Michael Faraday (England) 1845 described the rotation of the plane of polarized light that passed through glass in a magnetic field.


71

POLARIZATIONSOLO

Pockels Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318)

Vnr

nretardatio

Ern

n

n

Ln

Ln

yx

z

yy

xx

3

0''

32

''

''

2

:

21

2

2

λπ

λπ

λπ

=Φ−Φ=Φ

⇒

=∆−=

∆

=Φ

=Φ

Pockels Effect 1893

Electro – Optical Effects The electro – optical effects are called Pockels or Kerr where the refractive indexchanges linearly or quadraticly, respectively.

==

=

∆

Kerrk

PockelskEKn

nk

2

1102

Frederich Carl AlwinPockels

(1865-1913)

http://www.physi.uni-heidelberg.de/~schmiedm/Vorlesung/LasPhys02/LectureNotes/OpticsCrystals.pdf#search='Pockels%20Effect'

http://en.wikipedia.org/wiki/Pockels_effect

72

POLARIZATIONSOLO

Electro – Optical Effects

Kerr Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318)

The electro – optical effects are Kerr or Pockels

2

0 EKn λ=∆2

2

02

d

VKλπ=Φ∆

In the Kerr Electro-optic Effect 1875 is the electric field that causes the substance tobecome birefringent.


73

POLARIZATIONSOLO

Plane Polarized Wave in an Absorbing Medium

Circular Polarized Wave in an Absorbing Medium

a, 08/26/2005

1. Demos from http://www.enzim.hu/~szia/cddemo/ Andras Szilagyi

74

POLARIZATIONSOLO

Plane Polarized Wave in an Refracting Medium

Circular Polarized Wave in an Refracting Medium

a, 08/26/2005


75

POLARIZATIONSOLO

Plane Polarized Wave in a Medium Showing Circular Dichroism

Plane Polarized Wave in a Medium Showing Circular Birefrigens

a, 08/26/2005


76

POLARIZATIONSOLO

Plane Polarized Wave in a Medium Showing Both Circular Dichroism and Circular Birefrigens

a, 08/26/2005


77

OPTICSSOLO

http://microscopy.fsu.edu/


78

OPTICSSOLO

References Optics Polarization

A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984

M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980

E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8

C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996

G.R. Fowles, “Introduction to Modern Optics”,2nd Ed., Dover, 1975, Ch.2

M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986

http://en.wikipedia.org/wiki/Polarization

W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969

E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003

W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1, Dowden, Hutchinson & Ross, Inc., 1975


79


References Electromagnetics

J.D. Jackson, “Classical Electrodynamics”, 3rd Ed., John Wiley & Sons, 1999

R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966

J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941

W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison-Wesley, 1962

F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addison-Wesley, 1981

A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,John Wiley & Sons, 1988

80


References

1. W.K.H. Panofsky & M. Phillips, “Classical Electricity and Magnetism”,

2. J.D. Jackson, “Classical Electrodynamics”,

3. R.S. Elliott, “Electromagnetics”,

4. A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,


January 4, 2015 81

SOLO POLARIZATION

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –2013

Stanford University1983 – 1986 PhD AA

polarization

Science

unpolarized light

polarizationsolo light

light source

polrized light

scattered light

direction of propagation

propagation direction

principles of optics